I agree with all of that, and I think if you simulated it you’d indeed find a large bias for the number of small prime divisors to be even. The problem is 1/log(x)2 is such a small bias in expectation that it will already be offset by just the primes in the interval [x,2x] which contributes on the order of ∼1/logx to the average in the opposite direction, since all primes trivially have an odd number of divisors.
Furthermore, there are subtleties here about exactly how much independence you need. If you want everything to be jointly independent then you can really only work with the primes up to logx while being safe—this is because the product of the primes up to x is roughly of order ex. Once you go past that, while correlations involving only a small number of primes are still fine, correlations involving lots of primes break down, and for parity problems you need to control the entire joint distribution in a fine way.
This is not a problem for the normal approximation because to show convergence in distribution to a normal distribution, you just need to show all the moments converge to the right values and use a result like Stone-Weierstrass to approximate any continuous test function uniformly by a polynomial. You can do this just by working with primes up to xα for α depending on the exact moment you’re studying. However, this result is really “low resolution”, as you correctly identify.
I agree with all of that, and I think if you simulated it you’d indeed find a large bias for the number of small prime divisors to be even. The problem is 1/log(x)2 is such a small bias in expectation that it will already be offset by just the primes in the interval [x,2x] which contributes on the order of ∼1/logx to the average in the opposite direction, since all primes trivially have an odd number of divisors.
Furthermore, there are subtleties here about exactly how much independence you need. If you want everything to be jointly independent then you can really only work with the primes up to logx while being safe—this is because the product of the primes up to x is roughly of order ex. Once you go past that, while correlations involving only a small number of primes are still fine, correlations involving lots of primes break down, and for parity problems you need to control the entire joint distribution in a fine way.
This is not a problem for the normal approximation because to show convergence in distribution to a normal distribution, you just need to show all the moments converge to the right values and use a result like Stone-Weierstrass to approximate any continuous test function uniformly by a polynomial. You can do this just by working with primes up to xα for α depending on the exact moment you’re studying. However, this result is really “low resolution”, as you correctly identify.